Abstract
The following problem is considered: Given that the frequency distribution of the errors of measurement is known, determine or estimate the distribution of true scores from the distribution of observed scores for a group of examinees. Typically this problem does not have a unique solution. However, if the true-score distribution is “smooth,” then any two smooth solutions to the problem will differ little from each other. Methods for finding smooth solutions are developed a) for a population and b) for a sample of examinees. The results of a number of tryouts on actual test data are summarized.
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Aitkin, A. C., & Gonin, H. T. On fourfold sampling with and without replacement.Proceedings of the Royal Society of Edinburgh, 1935,55, 114–125.
Deely, J. J., & Kruse, R. L. Construction of sequences estimating the mixing distribution.The Annals of Mathematical Statistics, 1968,39, 286–288.
Eisenberg, H. B., Geoghagen, R. R. M., & Walsh, J. E. A general probability model for binomial events with application to surgical mortality.Biometrics, 1963,19, 152–157.
Jordan, C.Calculus of finite differences. (2nd ed.) New York: Chelsea, 1947.
Kendall, M. G., & Stuart, A.The advanced theory of statistics. New York: Hafner, 1958–61, 2 volumes.
Kenneth, P., & Taylor, G. E. Solution of variational problems with bounded control variables by means of the generalized Newton-Raphson method. In A. Lavi & T. Vogl (Eds.),Recent advances in optimization techniques. New York: Wiley, 1966.
Leitmann, G. Variational problems with bounded control variables. In G. Leitmann (Ed.),Optimization techniques. New York: Academic Press, 1962. Pp. 171–204.
Lord, F. M. A theory of test scores.Psychometric Monograph, 1952, No. 7.
Lord, F. M. A strong true-score theory, with applications.Psychometrika, 1965,30, 239–270.
Lord, F. M. Estimating item characteristic curves without knowledge of their mathematical form. ETS Research Bulletin 68-8 and ONR Technical Report, Contract Nonr-2752(00). Princeton, N. J.: Educational Testing Service, 1968.
Lord, F. M., & Lees, Dinan. Estimating true-score distributions for mental tests (Methods 12, 14, 15). ETS Research Memorandum 67-1 and ONR Technical Report, Contract Nonr-2752(00). Princeton, N. J.: Educational Testing Service, 1967. (a)
Lord, F. M., & Lees, Diana. Estimating true-score distributions for mental tests (Method 16). ETS Research Bulletin 67-7 and ONR Technical Report, Contract Nonr-2752(00). Princeton, N. J.: Educational Testing Service, 1967. (b)
Lord, F. M., & Novick, M. R.Statistical theories of mental test scores. Reading, Mass.: Addison-Wesley, 1968.
Maritz, J. S. Smooth empirical Bayes estimation for one-parameter discrete distributions.Biometrika, 1966,53, 417–429.
Pars, L. A.An introduction to the calculus of variations. London: Heinemann, 1962.
Phillips, D. L. A technique for the numerical solution of certain integral equations of the first kind.Journal of the Association for Computing Machinery, 1962,9, 84–97.
Rao, C. R.Linear statistical inference and its applications. New York: Wiley, 1965.
Riordan, J.An introduction to combinatorial analysis. New York: Wiley, 1958.
Robbins, H. An empirical Bayes approach to statistics.Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press, 1956, vol. 1, pp. 157–163.
Robbins, H. The empirical Bayes approach to testing statistical hypotheses.Review of the International Statistical Institute, 1963,31, (2), 195–208.
Robbins, H. The empirical Bayes approach to statistical decision problems.The Annals of Mathematical Statistics, 1964,35, 1–20.
Skellam, J. G. A probability distribution derived from the binomial distribution by regarding the probability of success as variable between sets of trials.Journal of the Royal Statistical Society, Series B, 1948,10, 257–261.
Tikhonov, A. N. Regularization of incorrectly posed problems.Soviet Mathematics Doklady, 1963,4, 1624–1627.
Tricomi, F. G.Integral equations. New York: Interscience, 1957.
Twomey, S. On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature.Journal of the Association for Computing Machinery, 1963,10, 97–101.
Walsh, J. E. Corrections to two papers concerned with binomial events.Sankhyā, 1963,25, 427.
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The writer wishes to thank Diana Lees and Virginia Lennon, who wrote the computer programs, carried out some of the mathematical derivations, and helped with other important aspects of the work. This work was supported in part by contract Nonr-2752(00) between the Office of Naval Research and Educational Testing Service. Reproduction, translation, use and disposal in whole or in part by or for the United States Government is permitted.
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Lord, F.M. Estimating true-score distributions in psychological testing (an empirical bayes estimation problem). Psychometrika 34, 259–299 (1969). https://doi.org/10.1007/BF02289358
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DOI: https://doi.org/10.1007/BF02289358